图的{P4}——分解

一个图G的路分解是指一路集合使得G的每条边恰好出现在其中一条路上．记Pl长度为l-1的路，如果G能够分解成若干个Pl，则称G存在{Pl}——分解，关于图的给定长路分解问题主要结果有：（i）连通图G存在{P3}-分解当且仅当G有偶数条边（见[1]）；（ii）连通图G存在{P3，P4}-分解当且仅当G不是C3和奇树，这里C3的长度为3的圈而奇树是所有顶点皆度数为奇数的树（见[3]）．本文讨论了3正则图的{P4}--分解情况，并构造证明了边数为3k（k∈Z且k≥2）的完全图Kn和完全二部图Kr,s存在{P4}-分解.     

仙人掌图的L（2，1）-标号

一个图G的L（2，1）-标号是给图G上的顶点分配非负整数标号，使得G上相邻的两个点的标号至少相差2，距离为2的两个点的标号则不同．G的L（2，1）-标号数λ（G）是所有能使图G正常标号的最小标号．如果一个图的任何两个圈不含有公共边，则称这个图为仙人掌图．显然树是它的一个子图类．对于任何树T，有△（T） ＋ 1 ≤λ（T） ≤△A（T）＋2．本文中我们证明了在一些条件下，这个界也适用于仙人掌图．     

基本圈与图的曲面嵌入

本文研究图的基本圈与图在可定向曲面上的嵌入之间的关系．本文结果表明：一个图G可以嵌入到亏格至少为g的可定向曲面上的充分必要条件是：对于G中任意一个支撑树T,存在一个基本圈序列C1,C2,…,Q2g,使得对于每一个i：1≤i≤g,C2i-1∩C2i≠0．特别地,在T的β（G）个基本圈中有基本圈序列C1,C2…,Q2γM（G）,使得Qt-1∩C2t≠0对于每一个i：1≤i≤γM（G）成立．这里β（G）和γM（G）分别是G的Betti数和最大可定向亏格．这个结果的意义在于：我们可以从任意一个支撑树（可以具有任意奇连通分支数）出发去构造图在可定向曲面上的嵌入．这在本质上有别于Xuong与Liu在最大亏格方面的工作（即,从具有最小奇连通分支数的支撑树出发构造图嵌入）．事实上,这个结果在本质上同时推广了Xuong-Liu与Fu等在最大亏格方面的工作．作为这一结果的直接应用,本文得到以下结果：（1）提出了用于计算图的最大亏格的新条件,它尤其适用于计算具有特定边割（edge—cut）图的最大亏格．并得到一些新的与已知的著名结果（包括Huang在曲面嵌入图方面的工作）．（2）最大亏格问题可以归结为在基本相交图中求最大对集问题．结合Micali-Vazirani的一个有效算法,我们设计出了一个用于计算图的最大亏格的多项式算法,它的复杂度是O（（β（G））^5/2）,这一算法与Furst等人的算法相比更加直接、便于计算．     

Mycielski图的BBC染色

基于图G的Mycielski图M（G）,研究xb（G,TG）与xb（M（G）,T’）之间的关系以及xb（G,TG）与xb（M（G）,T＂）之间的关系,其中Tc为G的生成树,T’,T＂分别为M（G）的两类特殊生成树．并给出当G为二部图,完全图以及Halin图时,Xb（M（G）,T＂）的值．     

图的{P_4}—分解

一个图G的路分解是指一路集合使得G的每条边恰好出现在其中一条路上.记Pl长度为l-1的路,如果G能够分解成若干个Pl,则称G存在{Pl}—分解.关于图的给定长路分解问题主要结果有:(i)连通图G存在{P3}—分解当且仅当G有偶数条边(见[1]);(ii)连通图G存在{P3,P4}—分解当且仅当G不是C3和奇树,这里C3的长度为3的圈而奇树是所有顶点皆度数为奇数的树(见[3]).本文讨论了3正则图的{P4}—分解情况,并构造证明了边数为3k(k热∈Z且k≥2)的完全图Kn和完全二部图Kr,s存在{P4}—分解.     

支撑树参数值的分布

本文通过一个基本定理证明了支撑树基本参数(但直径参数例外)值的分布是整数集Z的某个区间(即是连续整数集)。从而,使得G.Chartrand问题的肯定答案[2,3,4]成为该基本定理的推论。对例外的直径参数举了反例。本文所引用的记号均与一致,对[5]中没有的记号作如下的约定。记Ω(G)为G的支撑树全体所成之集,φ是Ω(G)到实数集R的映射,N_a(T)=|{v∈V(T)|d_r(v)≤α,α≥1}|为顶点v的度截尾数。设Z为正整数集,我们称[a,b]={x∈Z|a≤x≤b}为Z的区间(即为连续整数集)。对给定的支撑树T和余树边e,T+e中唯一圈称为基本圈,记为C_r(e)。     

没有5-圈的平面图的BB-染色

设H为G的一个生成子图,（G,H）的一个BB-k-染色是指一个映射f：V（G）→{1,2,…,k},当uv∈E（H）,｜f（u）-f（v）｜≥2;当uv∈E（G）／E（H）,｜f（u）-f（v）｜≥1.定义（G,H）的BB色数x_b（G,H）为最小的整数k,使得（G,H）是BB-k可染的.本文研究了对于任意的连通,非二部平面图G,且G没有5-圈,都存在一棵生成树T,使得x_b（G,T）=4.     

图的生成树, 基本圈与Betti亏数

G为图且T是G的一棵生成树. 记号ξ(G, T)表示G＼E(T)中边数为奇数的连通分支个数. 文献［2］称ξ(G)=min[DD(X]T[DD)]ξ(G, T)为图G的Betti亏数, 这里min取遍G的所有生成树T. 由文献［2］知, 确定一个图G的最大亏格主要确定这个图的Betii亏数ξ(G).该文研究与Betti亏数有关的图的特征结构, 得到了关于图的最大亏格的若干结果.     

字典序地生成有序根树的一个算法

§1.引言由于树的生成在计算机科学中有着重要应用,近年来许多文章研究了树的生成,其中大多数文章是讨论2分树及 k 分树的生成.研究一般有序根树的文章尚少.文献[1]给出了有序根树的一个序列表示法,并描述了一个生成有序根树的算法.文献[2]及[3]讨论了生成2分树及 k 分树的算法.本文用0,1序列表示有序根树,并给出了一个字典序地生成具有 n 个顶点的所有有序根树的算法.本文的表示法及算法与文献[1]中所提方法不同.本算法亦可用来生成具有 n 个叶子的所有2分树.它比[2]中的算法更简单.本文中未加说明的术语皆见[1].     

关于图的余树的奇连通分支数的内插定理

本文研究了连通图的余树的奇连通分支数与其可定向嵌入的关系.我们先给出了关于连通图的余树的奇连通分支数的内插定理.作为其应用,我们推广了Xuong和刘彦佩关于图的最大亏格的计算公式,并且证明了如下结果:任意一个连通图G一定满足下列条件之一: (a)对于任意的满足γ(G)≤g≤γM(G)整数g,只要图G嵌入到可定向曲面Sg上,就存在支撑树T,使g-1/2β(G)-ω(T)),其中,γ(G)与γM(G)分别是图G的最小和最大亏格,β(G)与ω(T)分别是图G的Betti数和由T确定的余树的奇连通分支数; (b)对连通图G的任意一个支撑树T,G可以嵌入某个可定向曲面上使其恰好有ω(T) 1个面.特别地,我们给出了所有非平面的3-正则的Hamilton图G所嵌入的可定向曲面的亏格的计算公式.     

Pendants-median支撑树及其一个相关问题:复杂性和算法

The complexity status of Pendants-median spanning tree problem is an open problem. Using the complexity of the X3C problem, the paper proves that Pendants-median spanning tree problem is NP-complete. Global-median spanning tree problem is a related problem. Using the complexity of 3SAT, the paper proves that this problem is also NP-complete, and a polynomial -time algorithm to this problem is given, whose time complexity is O(n^3).     

Greedy heuristics for the diameter-constrained minimum spanning tree problem

The diameter-constrained minimum spanning tree problem is an NP-hard combinatorial optimization problem that seeks a minimum cost spanning tree with a limit D imposed upon the length of any path in the tree. We begin by presenting four constructive greedy heuristics and, secondly, we present some second-order heuristics, performing some improvements on feasible solutions, hopefully leading to better objective function values. We present a heuristic with an edge exchange mechanism, another that transforms a feasible spanning tree solution into a feasible diameter-constrained spanning tree solution, and finally another with a repetitive mechanism. Computational results show that repetitive heuristics can improve considerably over the results of the greedy constructive heuristics, but using a huge amount of computation time. To obtain computational results, we use instances of the problem corresponding to complete graphs with a number of nodes between 20 and 60 and with the value of D varying between 4 and 9.     

Minimum Global Height支撑树及相关问题

本文研究了两个组合优化问题:minimum g1obal height支撑树和minimum aveageheight支撑树问题.利用3SAT问题的时间复杂性,本文证明了这两个问题都是NP-hard的,并分别给出了一个算法,即（mgh）-算法和（mah）-算法.在非负网络中,这两个算法的时间复杂性都为O（n3）.利用第一个问题的复杂性,本文证明了minimum height支撑树问题也是NP-hard的,从而纠正了有关文献中的一个错误结论.     

A CAPACITY EXPANSION PROBLEM WITH BUDGET CONSTRAINT AND BOTTLENECK LIMITATION

1 IntroductionIn [2] tl1e authors considered a type of coustrained maximim capacity path problem whichcan be described briefly as: kuowing the costs for expallding one unit of capacity along differentedge8 of a l1etwork aud the availabIe budget, how to iucrease the caparities of the edges so thattlle capasity between any pair of nodes in the lletwork can be raised unifornily to the maximumextent? As the total cost is a summation of the expansion costs on all edges, this problem i8related to mi…     

Minimum congestion spanning trees in bipartite and random graphs

The first problem considered in this article reads: is it possible to find upper estimates for the spanning tree congestion in bipartite graphs, which are better than those for general graphs? It is proved that there exists a bipartite version of the known graph with spanning tree congestion of order n 3 2 , where n is the number of vertices. The second problem is to estimate spanning tree congestion of random graphs. It is proved that the standard model of random graphs cannot be used to find graphs whose spanning tree congestion has order greater than n 3 2 .     

A Polynomial Time Approximation Scheme for Optimal Product-Requirement Communication Spanning Trees

Given an undirected graph with nonnegative edge lengths and nonnegative vertex weights, the routing requirement of a pair of vertices is assumed to be the product of their weights. The routing cost for a pair of vertices on a given spanning tree is defined as the length of the path between them multiplied by their routing requirement. The optimal product-requirement communication spanning tree is the spanning tree with minimum total routing cost summed over all pairs of vertices. This problem arises in network design and computational biology. For the special case that all vertex weights are identical, it has been shown that the problem is NP-hard and that there is a polynomial time approximation scheme for it. In this paper we show that the generalized problem also admits a polynomial time approximation scheme.     

On Minimum Edge Ranking Spanning Trees

In this paper, we introduce the problem of computing a minimum edge ranking spanning tree (MERST); i.e., find a spanning tree of a given graph G whose edge ranking is minimum. Although the minimum edge ranking of a given tree can be computed in polynomial time, we show that problem MERST is NP-hard. Furthermore, we present an approximation algorithm for MERST, which realizes its worst case performance ratio where n is the number of vertices in G and Δ* is the maximum degree of a spanning tree whose maximum degree is minimum. Although the approximation algorithm is a combination of two existing algorithms for the restricted spanning tree problem and for the minimum edge ranking problem of trees, the analysis is based on novel properties of the edge ranking of trees.     

Paths, trees and matchings under disjunctive constraints

We study the minimum spanning tree problem, the maximum matching problem and the shortest path problem subject to binary disjunctive constraints: A negative disjunctive constraint states that a certain pair of edges cannot be contained simultaneously in a feasible solution. It is convenient to represent these negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the edges of the underlying graph, and whose edges encode the constraints.We prove that the minimum spanning tree problem is strongly NP-hard, even if every connected component of the conflict graph is a path of length two. On the positive side, this problem is polynomially solvable if every connected component is a single edge (that is, a path of length one). The maximum matching problem is NP-hard for conflict graphs where every connected component is a single edge.Furthermore we will also investigate these graph problems under positive disjunctive constraints: In this setting for certain pairs of edges, a feasible solution must contain at least one edge from every pair. We establish a number of complexity results for these variants including APX-hardness for the shortest path problem.     

Greedy Randomized Adaptive Search and Variable Neighbourhood Search for the minimum labelling spanning tree problem

This paper studies heuristics for the minimum labelling spanning tree (MLST) problem. The purpose is to find a spanning tree using edges that are as similar as possible. Given an undirected labelled connected graph, the minimum labelling spanning tree problem seeks a spanning tree whose edges have the smallest number of distinct labels. This problem has been shown to be NP-hard. A Greedy Randomized Adaptive Search Procedure (GRASP) and a Variable Neighbourhood Search (VNS) are proposed in this paper. They are compared with other algorithms recommended in the literature: the Modified Genetic Algorithm and the Pilot Method. Nonparametric statistical tests show that the heuristics based on GRASP and VNS outperform the other algorithms tested. Furthermore, a comparison with the results provided by an exact approach shows that we may quickly obtain optimal or near-optimal solutions with the proposed heuristics.     

A Benders decomposition approach for the robust spanning tree problem with interval data

The robust spanning tree problem is a variation, motivated by telecommunications applications, of the classic minimum spanning tree problem. In the robust spanning tree problem edge costs lie in an interval instead of having a fixed value.Interval numbers model uncertainty about the exact cost values. A robust spanning tree is a spanning tree whose total cost minimizes the maximum deviation from the optimal spanning tree over all realizations of the edge costs. This robustness concept is formalized in mathematical terms and is used to drive optimization.This paper describes a new exact method, based on Benders decomposition, for the robust spanning tree problem with interval data. Computational results highlight the efficiency of the new method, which is shown to be very fast on all the benchmarks considered, and in particular on those that were harder to solve for the methods previously known.